Remarks on the quadratic Hessian equation

We prove that viscosity solutions to the quadratic Hessian equation $$\sigma_2(D^2u) = 1$$ cannot touch a harmonic function on a minimal surface from below. This can be viewed as a form of strict $2$-convexity. We also prove an a priori interior $C^2$ estimate in terms of the $W^{2,\,p}$ norm, for any $p > 2$. Finally, we discuss how these results rule out certain strategies for constructing counterexamples to regularity.